Factorization results for left polynomials in some associative real algebras: state of the art, applications, and open questions. (English) Zbl 1425.12001

Li, Zijia; Scharler, Daniel F.; Schröcker, Hans-Peter

J. Comput. Appl. Math. 349, 508-522 (2019).

This paper is suited for graduate students and researchers of hypercomplex algebra applied to advanced mechanical problems. The origins of this work date back to the middle of the last (20th) century. It gradually evolved from motion problems described by quaternion (isomorphic to the even subalgebras of \(\mathrm{Cl}(3,0)\) or \(\mathrm{Cl}(0,3)\)) polynomials to the use of dual quaternions (with an extra commutative unit squaring to zero, isomorphic to the even subalgebra of \(\mathrm{Cl}(3,0,1)\)), split quaternions (isomorphic to the even subalgebra of \(\mathrm{Cl}(1,2)\)), and Clifford algebras in general.
Section 2 reviews the subject of polynomial factorizations of (non-commutative) rings, including an algorithm (Alg. 1), that allows the computation of linear right factors from zeros. Section 3 is on the existence (Th. 3) of polynomial factorizations over quaternions (or more general finite-dimensional associative real involutive algebras), key roles being played by the so-called maximal real polynomial factor (being equal to one) and the involution. Based on this theorem, the second algorithm (Alg. 2) produces factorizations.
Additionally, Section 4 provides examples of polynomial factorizations by first introducing Clifford algebras and spin groups based on these algebras. Of particular interest are Clifford algebras that allow isomorphisms to (non-)Euclidean space transformation groups, because factorization there can decompose rational motions into elementary motion products. In particular, here we find brief characterizations of quaternions, split quaternions and dual quaternions, and a range of characteristic examples (some of which conclude that no factorizations exist) for polynomial factorizations involving these algebras.
Section 5 is on applications of factorization in mechanism science, including concrete examples of such mechanisms (anti-parallelogram mechanism in Euclidean and hyperbolic geometry, parallelogram linkage and a scissor linkage). Finally, Section 6 goes beyond the previous theory showing more results and examples regarding the wider applicability of Alg. 2 (with example), the factorization of quadratic split quaternion polynomials, factorization of non-generic motion polynomials, of unbounded motion polynomials, and by using a projection method (for non-motion polynomials in dual quaternions, with a generalization to polynomials in Clifford algebras \(\mathrm{Cl}(p,q,1)\)).

Reviewer: Eckhard Hitzer (Tokyo)

MSC:

12D05	Polynomials in real and complex fields: factorization
15A66	Clifford algebras, spinors
16S36	Ordinary and skew polynomial rings and semigroup rings
30C15	Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
70B15	Kinematics of mechanisms and robots

Keywords:

split quaternion; Clifford algebras; kinematics; quaternion; mechanism science; dual quaternion; polynomial division

References:

[1]	Niven, I., Equations in quaternions, Amer. Math. Monthly, 48, 10, 654-661, (1941) · Zbl 0060.08002
[2]	Gordon, B.; Motzkin, T. S., On the zeros of polynomials over division rings, Trans. Amer. Math. Soc., 116, 218-226, (1965) · Zbl 0141.03002
[3]	Hegedüs, G.; Schicho, J.; Schröcker, H. P., Factorization of rational curves in the study quadric and revolute linkages, Mech. Machine Theory, 69, 1, 142-152, (2013), http://arxiv.org/abs/1202.0139
[4]	Z. Li, T.D. Rad, J. Schicho, H.P. Schröcker, Factorization of rational motions: A survey with examples and applications, in: Chang, S.-H. (Ed.), Proceedings of the 14th IFToMM World Congress, 2015.
[5]	Z. Li, J. Schicho, H.P. Schröcker, 7R Darboux linkages by factorization of motion polynomials, in: Chang, S.-H. (Ed.), Proceedings of the 14th IFToMM World Congress, 2015.
[6]	Li, Z.; Schicho, J.; Schröcker, H. P., Spatial straight-line linkages by factorization of motion polynomials, J. Mech. Robotics, 8, 2, 021002, (2016)
[7]	Gallet, M.; Koutschan, C.; Li, Z.; Regensburger, G.; Schicho, J.; Villamizar, N., Planar linkages following a prescribed motion, Math. Comp., 87, 473-506, (2017) · Zbl 1404.70007
[8]	Z. Li, J. Schicho, H.P. Schröcker, Kempe’s universality theorem for rational space curves, Found. Comput. Math., http://arxiv.org/abs/1509.08690. · Zbl 1430.70006
[9]	Scharler, D., Characterization of Lines in the Extended Kinematic Image Space, (2017), University of Innsbruck
[10]	Li, Z.; Schicho, J.; Schröcker, H. P., Factorization of Motion Polynomials, J. Symb. Comput., (2018)
[11]	Rad, T. D., Factorization of Motion Polynomials and its Application in Mechanism Science, (2018), University of Innsbruck
[12]	Ore, O., Theory of non-commutative polynomials, Annh. of Math. (2), 34, 3, 480-508, (1933) · Zbl 0007.15101
[13]	Palais, R. S., The classification of real division algebras, Amer. Math. Monthly, 75, 4, 366-368, (1968) · Zbl 0159.04403
[14]	Corless, R. M.; Watt, S. M.; Zhi, L., QR factoring to compute the GCD of univariate approximate polynomials, IEEE Trans. Signal Process., 52, 12, 3394-3402, (2004) · Zbl 1372.65120
[15]	Kaltofen, E.; Yang, Z.; Zhi, L., Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials, (Dumas, J. G., Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC ’06, (2006), ACM: ACM Genoa, Italy), 169-176 · Zbl 1356.12011
[16]	Bini, D.; Pan, V. Y., Polynomial and Matrix Computations: Fundamental Algorithms, (2012), Springer Science & Business Media
[17]	Pan, V. Y., Computation of approximate polynomial GCDs and an extension, Inf. Comput., 167, 2, 71-85, (2001) · Zbl 1005.12004
[18]	Li, Z.; Yang, Z.; Zhi, L., Blind image deconvolution via fast approximate GCD, (Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, (2010), ACM), 155-162 · Zbl 1321.68442
[19]	Klawitter, D., Clifford Algebras. Geometric Modelling and Chain Geometries with Application in Kinematics, (2015), Springer Spektrum · Zbl 1310.15037
[20]	Selig, J., Geometric fundamentals of robotics, (Monographs in Computer Science, (2005), Springer) · Zbl 1062.93002
[21]	Pottmann, H.; Wallner, J., Computational line geometry, (Mathematics and Visualization, (2010), Springer), 2nd printing · Zbl 1175.51014
[22]	Wildberger, N., Universal hyperbolic geometry I: Trigonometry, Geom. Dedicata, 163, 215-274, (2013) · Zbl 1277.51018
[23]	Li, Z.; Schicho, J.; Schröcker, H. P., The geometry of quadratic quaternion polynomials in Euclidean and Non-Euclidean Planes, (Cocchiarella, L., ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics, (2019), Springer International Publishing: Springer International Publishing Cham), 298-309, http://arxiv.org/abs/1805.03539 · Zbl 1400.51009
[24]	Hegedüs, G.; Schicho, J.; Schröcker, H. P., Four-Pose synthesis of angle-symmetric 6R linkages, J. Mech. Robotics, 7, 4, (2015)